$\"\"$

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The equations of the curves are $\"\"$ and $\"\"$ .

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Determine the points of intersection by plotting their graphs.

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Graph :

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(1). Draw the coordinate plane.

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(2). Graph the curves $\"\"$ and $\"\"$ .

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$\"\"$

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Observe the graph, the two curves intersect at the points $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$ .

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Consider $\"\"$ and $\"\"$ .

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$\"\"$

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Apply the formula : $\"\"$ .

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$\"\"$

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Apply integration :

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$\"\"$

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$\"\"$ .

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$\"\"$

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Consider $\"\"$ and $\"\"$ .

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Moment about $\"\"$ - axis $\"\"$ of a planar lamina is defined as,

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$\"\"$ .

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Where, $\"\"$ is Uniform density of a planar lamina.

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Here $\"\"$ value varies form $\"\"$ to $\"\"$ , so $\"\"$ and $\"\"$ .

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Substitute $\"\"$ , $\"\"$ and limits of integral in the above formula.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Moment about $\"\"$ - axis $\"\"$ of a planar lamina is defined as,

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$\"\"$ .

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Where, $\"\"$ is Uniform density of a planar lamina.

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Here $\"\"$ value varies form $\"\"$ to $\"\"$ , so $\"\"$ and $\"\"$ .

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Substitute $\"\"$ , $\"\"$ and limits of integral in the above formula.

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$\"\"$

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$\"\"$

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Apply the formula : $\"\"$ .

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$\"\"$

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Apply integration :

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$\"\"$

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$\"\"$ .

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$\"\"$

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Find the values of $\"\"$ :

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Calculate $\"\"$ :

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$\"\"$

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Calculate $\"\"$ :

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$\"\"$

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$\"\"$

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Moments $\"\"$ and center of mass $\"\"$ .